A Hilbert Lemniscate Theorem in \mathbb{C}^2

Special polyhedra for Reinhardt domains

“…When n = 1, its solution (D. Hilbert, 1897) is known as the Hilbert Lemniscate theorem; a proof can be found, e.g., in [6]. For n = 2, the problem was solved in [3] in the case of a circled set K (that is, ζK ⊆ K for any ζ from the closed unit disk), and the components of the mapping P can be chosen as homogeneous polynomials of equal degree with unique common zero at the origin.…”

“…The above results in [3], [4], [5] make use of approximation of the pluricomplex Green function for a compact set with pole at infinity by logarithms of moduli of equidimensional polynomial mappings. Our approach is based on approximation of pluricomplex Green functions for bounded Reinhardt domains with pole at the origin.…”

On Lelong-Bremermann Lemma

“…First we consider two lemmas. During the preparation of this paper for publication, we became aware of a result (see, preprint [3], Theorem 1.2) that is somehow stronger than Lemma 1 below. We decided to keep our proof of this lemma since it is more direct and does not use the Yulmukhamedov Lemma (see, e.g., [3], Lemma A).…”